I read a graphic novel called "Logicomix" years ago and have wondered this ever since.
There seems to be no established explanation of Wittgenstein's rhinoceros, just speculations. I may as well add one of my own :) (I don't know whether the following explanation has been offered before)
Wittgenstein was reported to have asserted, at the incident, that "nothing empirical is knowable". To me this immediately reminded one specific item of philosophy: David Hume and the problem of induction. Where else, in the history of philosophy, we have such a prominent argument against empirical knowledge in particular? Hume famously and convincingly argued, that all empirical knowledge was based on induction, and that induction had no rational basis. Induction had a merely psychological basis, in the psychology of habit. Hence Wittgenstein's "nothing empirical is knowable".
But how, you might ask yourself, is the problem of induction related to Wittgenstein? Well, it's rather straightforward really. Wittgenstein is endorsing Hume's position on induction (and on causation) in the Tractatus. This happens in the somewhat wild sixth part of the Tractatus, where Wittgenstein applies the philosophical view, which he develops in the previous parts, to a host of philosophical issues. One of his theses there is that there is no necessity except logical necessity. Here is how it goes:
6.362 What can be described can happen too: and what the law of causality is meant to exclude cannot even be described.
6.363 The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences.
6.3631 This procedure, however, has no logical justification but only a psychological one. It is clear that there are no grounds for believing that the simplest eventuality will in fact be realized.
6.36311 It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise.
6.37 There is no compulsion making one thing happen because another has happened. The only necessity that exists is logical necessity.
This view of Wittgenstein's will provide us with an easy explanation to his rhinoceros assertion. For just as we cannot know, according to Hume and Wittgenstein, whether the sun will rise tomorrow, or not, we cannot know whether there was a rhinoceros on Russell's room, or not. When e.g. Russell looked under one table, it is perfectly possible (that is, logically possible) that the rhinoceros popped up under the other table, and vice versa. The rhinoceros could have been on the ceiling, it could have been very small, it could have been invisible, etc. There would be no way to know otherwise. And so the riddle may be solved.
While Ram Tobolski clearly makes an important point. I think we can generalize even a bit further. For example what if we were looking into a sufficiently small and empty room from an outside window in such a way that we could see the entirety of it at once. The non-observation of the Rhino is still an empirical truth. Which Wittgenstein categorically denies. I imagine that further dialogue would go something like this:
You: "Well there you have it, there's no rhino in that room?"
W:"How do you know there is no Rhino in that room?"
You: "What? That room is only 4 x 2 meters! Rhino's are almost as big and hard to miss!"
W:"Are all Rhino's the same size?"
You: "Well, no but since it doesn't matter what claim we are making, as long as it's empirical we can just change our claim. Right?
W: Of course.
You: Well then we can certainly claim there are no Rhino's of 3 meters or more in that room!"
W:"How do you know, that there isn't a 3m long rhino in that room but you simply can not perceive it?"
and so on...
The way I see it, what Wittgenstein is pointing out is that eventually we have to ask 'What do we mean by "no rhino in the room"?' and the only way to answer that question without appealing to empirical facts is to define it by trivially true logical truths such as...
- What we mean by a rhino is at least 1m long.
- This room is less than 1m long.
- Therefore this room can contain no rhinos.
Wittgenstein's point was that the phrase "there is no rhinoceros in the room" was nonsensical and was not able to assert something meaningful about the world, while Russell saw it as some sort of truth. Why Wittgenstein refused to admit that there really was no rhinoceros in the room, we will never know... maybe to annoy Russell?